Basic intuition behind proving Runge's Theorem I am studying functional analysis and have a little bit of previous study in complex analysis. I want to prove Runge's theorem, but am finding it difficult to find a source that helps me understand exactly what is happening (especially due to the fact that many sources use different methods and different terminology).
Runge's Theorem (The version I am interested in) States that one can approximate a function which is holomorphic on
a compact set in $\mathbb{C}$ by rational functions with only one pole at a prescribed point in the complement of
the compact set. 
I have seen that a Corollary of the Hahn Banach Theorem is very useful along with the Riesz Representation theorem.
H.B : Conway (III) 6.13 :If N is a normed space and M is a linear subspace of N then,
Closure(M) = $\{ Ker f : f \in \bf{N'} \text{(dual space)}  \ \cap \ \ \bf{M}\subseteq Ker f \}$
For a locally compact topological space X let,
\begin{equation}
C_{0} (\bf{X}) = \{ f \in C(\bf{X},\mathbb{C}) \big | \forall \epsilon>0 : f^{-1} (\mathbb{C}\backslash B_{\epsilon}(0))  \ \ \text{Compact} \}
\end{equation}
(In Short) The Riesz Representation Theorem gives an interpretation of the dual $ C_{0} (\bf{X})'$
in terms of certain measures
on X.
Riesz Representation Theorem (Appendix C) Conway : 
If X is a locally compact space and $ \mu \in $ The measures of X, define,$F_{\mu} :  C_{0} (\bf{X}) \rightarrow \mathbb{C}$ by,
$F_{\mu} (f) = \int f d \mu $
Then,
$F_{\mu} \in  C_{0} (\bf{X})'$ and the map $\mu \rightarrow F_{\mu}$ is an isometric isomorphism of the measures of X onto $ C_{0} (\bf{X})'$.
I am confused with the notation. Here is a link to Conways book.http://www.faculty.jacobs-university.de/poswald/teaching/FunctAnal/handouts/John%20B.%20Conway%20A%20course%20in%20functional%20analysis%20%201997.pdf. Chapter (III) He calls $K$ the compact subset and $E$ the subset of $\mathbb{C}\backslash K$ He then  calls $R(K,E)$ the closure in the space $C(K)$ of the rational functions with poles in $E$.
Where I am stuck is understanding how he can say that (from the theorems I introduced above)all we need to prove Runge's theorem is  that,
if $\mu$ is a measure in $K$ and $\int g d \mu = 0$ for all $g \in R(K,E) $
then $\int f d \mu =0$.
I can follow the rest of the math in the proof in Conway but this is the key to the proof and I am finding it hard to understand. Can we relate this to $C_0(X)$?.The more books I look at the more notation I see, so I am just going to stick with this.
 A: Since $K$ is compact, $C_0(K)=C(K)$. Let $A$ be a Banach space and let $B$ be a subspace. Then $B$ is dense in $A$ if and only if every linear functional $\lambda$ on $A$ such that $\lambda(b)=0$ for all $b\in B$ is the zero functional on $A$. We apply this to the Banach space $A$ consisting of all functions which are uniform limits of functions holomorphic on $K$ and the subspace $R(K, E)$.
Disclaimer: I typed this on my phone, I will double check it later.
Edit (Later checking): The first result you quote (under the statement of Runge's theorem) is what I describe above.
Since $C_0(K)=C(K)$, and $A$ (as above) is a closed linear subspace (actually it is Banach algebra) of $C(K)$, it follows from the Riesz representation theorem that every linear functional on $A$ is represented by a measure on $K$. By the first result, it is enough to show that if $F_\mu$ (in your notation) is such that $F_\mu(g)=0$ for all $g\in R(X,E)$, where $E$ is chosen such that $\mathbb C\setminus E$ meets each component of $\mathbb C\setminus K$ at exactly one point, satisfies $F_\mu=0$. We must be careful here: this functional $F_\mu$ is not identically zero on $C(K)$ in general! (Not every continuous function on an arbirtary compact subset $X$ of $\mathbb C$ can be uniformly approximated by functions which are holomorphic on a neighbourhood of $X$.)
